Tutorial 6 The Trigonometric Functions MT129 – Calculus and Probability

Outline Radian Measure of Angles The Sine and the Cosine Differentiation and Integration of sine and cosine The Tangent and Other Trigonometric Functions MT129 – Calculus and Probability

Radians and Degrees MT129 – Calculus and Probability

Positive & Negative Angles Definition Example Positive Angle: An angle measured in the counter-clockwise direction Definition Example Negative Angle: An angle measured in the clockwise direction MT129 – Calculus and Probability

Converting Degrees to Radians EXAMPLE Convert the following to radian measure: SOLUTION MT129 – Calculus and Probability

Determining an Angle Give the radian measure of the angle described. EXAMPLE Give the radian measure of the angle described. SOLUTION The angle above consists of one full revolution (2π radians) plus one half-revolutions (π radians). Also, the angle is clockwise and therefore negative. That is, MT129 – Calculus and Probability

Sine & Cosine in Right Angle MT129 – Calculus and Probability

Sine & Cosine in a Unit Circle MT129 – Calculus and Probability

Properties of Sine & Cosine MT129 – Calculus and Probability

Calculating Sine & Cosine EXAMPLE Give the values of sin t and cos t, where t is the radian measure of the angle shown. SOLUTION We can immediately determine sin t. Since sin2t + cos2t = 1, we have Replace sin2t with (1/4)2. Take the square root of both sides. MT129 – Calculus and Probability

Using Sine & Cosine If t = 0.4 and a = 10, find c. EXAMPLE If t = 0.4 and a = 10, find c. SOLUTION Since cos(0.4) = 10/c, we get MT129 – Calculus and Probability

Determining an Angle t EXAMPLE Find t such that –π/2 ≤ t ≤ π/2 and t satisfies the stated condition SOLUTION One of our properties of sine is sin(–t) = –sin(t). And since –sin(3π/8) = sin(–3π/8) and –π/2 ≤ –3π/8 ≤ π/2, we have t = –3π/8. MT129 – Calculus and Probability

The Graphs of Sine & Cosine MT129 – Calculus and Probability

Derivatives of Sine & Cosine MT129 – Calculus and Probability

Differentiating Sine & Cosine EXAMPLE Differentiate the following: SOLUTION MT129 – Calculus and Probability

Differentiating Cosine in Application EXAMPLE Suppose that a person’s blood pressure P at time t (in seconds) is given by P = 100 + 20cos 6t. Find the maximum value of P (called the systolic pressure) and the minimum value of P (called the diastolic pressure) and give one or two values of t where these maximum and minimum values of P occur. SOLUTION The maximum value of P and the minimum value of P will occur where the function has relative minima and maxima. These relative extrema occur where the value of the first derivative is zero. This is the given function. Differentiate. Set P΄ equal to 0. Divide by -120. MT129 – Calculus and Probability

Differentiating Cosine in Application CONTINUED Notice that sin6t = 0 when 6t = 0, π, 2π, 3π,... . That is, when t = 0, π/6, π/3, π/2, ... . Now we can evaluate the original function at these values for t. t 100 + 20cos6t 120 π/6 80 π/3 π/2 Notice that the values of the function P cycle between 120 and 80. Therefore, the maximum value of the function is 120 and the minimum value is 80. MT129 – Calculus and Probability

Application of Differentiating & Integrating Sine EXAMPLE The average weekly temperature in Washington, D.C. t weeks after the beginning of the year is The graph of this function is sketched below. (a) What is the average weekly temperature at week 18? (b) At week 20, how fast is the temperature changing? MT129 – Calculus and Probability

Application of Differentiating & Integrating Sine SOLUTION (a) The time interval up to week 18 corresponds to t = 0 to t = 18. The average value of f (t) over this interval is MT129 – Calculus and Probability

Application of Differentiating & Integrating Sine CONTINUED Therefore, the average value of f (t) is about 47.359 degrees. (b) To determine how fast the temperature is changing at week 20, we need to evaluate f ΄(20). This is the given function. Differentiate. Simplify. Evaluate f ΄(20). Therefore, the temperature is changing at a rate of 1.579 degrees per week. MT129 – Calculus and Probability

Other Trigonometric Identities MT129 – Calculus and Probability

Applications of Tangent EXAMPLE Find the width of a river at points A and B if the angle BAC is 90°, the angle ACB is 40°, and the distance from A to C is 75 feet. r SOLUTION Let r denote the width of the river. Then equation (3) implies that We convert 40° into radians. We find that 40° = (π/180)40 radians ≈ 0.7 radians, and tan(0.7) ≈ 0.84229. Hence MT129 – Calculus and Probability

Tangent MT129 – Calculus and Probability

Differentiating Tangent EXAMPLE Differentiate. SOLUTION We find that MT129 – Calculus and Probability

Finding Anti-derivatives EXAMPLE Determine the following SOLUTION Using the rules of indefinite integrals, we have MT129 – Calculus and Probability Goldstein/SCHNIEDER/LAY, CALCULUS AND ITS APPLICATIONS, 11e – Slide #10